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May  2011, 10(3): 917-936. doi: 10.3934/cpaa.2011.10.917

Feedback control via inertial manifolds for nonautonomous evolution equations

Norbert Koksch 1, and  Stefan Siegmund 2,

1. 

Fachrichtung Mathematik, Technische Universitat Dresden, 01062 Dresden, Germany
2. 

Department of Mathematics, Dresden University of Technology, 01062 Dresden

Received  April 2009 Revised  September 2009 Published  December 2010

In this paper we extend a method to control the dynamics of evolution equations by finite dimensional controllers which was suggested by Brunovsky [3] to nonautonomous evolution equations using nonautonomous inertial manifold theory.
Keywords: feed-back control, process., Inertial manifold, nonautonomous dynamical systems.
Mathematics Subject Classification: Primary: 34C30; Secondary: 34D45, 34G20, 35B42, 37L2.
Citation: Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure and Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917
References:
[1]

L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Syst., 13 (1998), 265-280.

[2]

L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: doi:10.1016/S0362-546X(97)00569-5.

[3]

P. Brunovsky, Controlling the dynamics of scalar reaction diffusion equations by finite dimensional controllers, Proc. IFIP-IIASA Conf., Laxenburg/Austria 1989, Lect. Notes Control Inf. Sci. 154, 22-27 (1991).

[4]

S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317. doi: doi:10.1016/0022-0396(88)90007-1.

[5]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Differ. Equations, 13 (2001), 355-380. doi: doi:10.1023/A:1016684108862.

[6]

C. M. Dafermos, An invariance principle for compact processes, J. Differ. Equations, 9 (1971), 239-252; Erratum. Ibid. 10, 179-180.

[7]

C. M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory, 8 (1975), 142-149. doi: doi:10.1007/BF01762184.

[8]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Application," Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientific & Technical, Harlow, 1992.

[9]

J. Eisenfeld and V. Lakshmikantham, Comparison principle and nonlinear contractions in abstract spaces, J. Math. Anal. Appl., 49 (1975), 504-511. doi: doi:10.1016/0022-247X(75)90193-6.

[10]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353.

[11]

C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Differ. Equations, 1 (1989), 199-244. doi: doi:10.1007/BF01047831.

[12]

C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Varieties inertielles pour l'equation de Kuramoto-Sivashinsky. (Inertial manifolds for the Kuramoto-Sivashinsky equation), C. R. Acad. Sci., Paris, Ser. I, 301 (1985), 285-288.

[13]

A. Yu. Goritskiĭ and V. V. Chepyzhov, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 485-511; translation from Mat. Sb., 196 (2005), 23-50.

[14]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, vol. 850, Springer, 1981.

[15]

N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equations, 14 (2002), 889-941. doi: doi:10.1023/A:1020768711975.

[16]

M. A. Krasnoselski, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems, the Method of Positive Operators," Heldermann Verlag, Berlin, 1989.

[17]

Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems, 5 (1999), 233-268. doi: doi:10.3934/dcds.1999.5.233.

[18]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995.

[19]

L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, pp. 161-168, 1987.

[20]

J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866.

[21]

A. V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47. doi: doi:10.1070/IM1994v043n01ABEH001557.

[22]

R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dyn. Differ. Equations, 15 (2003), 61-86. doi: doi:10.1023/A:1026153311546.

[23]

H. Sano and N. Kunimatsu, Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems, IMA J. Math. Control Inform., 11 (1994), 75-92. doi: doi:10.1093/imamci/11.1.75.

[24]

H. Sano and N. Kunimatsu, An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems, J. Math. Anal. Appl., 196 (1995), 18-42. doi: doi:10.1006/jmaa.1995.1396.

[25]

G. R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Differ. Equations, 96 (1992), 203-255. doi: doi:10.1016/0022-0396(92)90152-D.

[26]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer, New York, 2002.

[27]

S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziaris, Order reduction for nonlinear dynamic models of distributed reacting systems, Journal of Process Control, 10 (2000), 177-184. doi: doi:10.1016/S0959-1524(99)00029-3.

[28]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997.

show all references

References:
[1]

L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Syst., 13 (1998), 265-280.

[2]

L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: doi:10.1016/S0362-546X(97)00569-5.

[3]

P. Brunovsky, Controlling the dynamics of scalar reaction diffusion equations by finite dimensional controllers, Proc. IFIP-IIASA Conf., Laxenburg/Austria 1989, Lect. Notes Control Inf. Sci. 154, 22-27 (1991).

[4]

S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317. doi: doi:10.1016/0022-0396(88)90007-1.

[5]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Differ. Equations, 13 (2001), 355-380. doi: doi:10.1023/A:1016684108862.

[6]

C. M. Dafermos, An invariance principle for compact processes, J. Differ. Equations, 9 (1971), 239-252; Erratum. Ibid. 10, 179-180.

[7]

C. M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory, 8 (1975), 142-149. doi: doi:10.1007/BF01762184.

[8]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Application," Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientific & Technical, Harlow, 1992.

[9]

J. Eisenfeld and V. Lakshmikantham, Comparison principle and nonlinear contractions in abstract spaces, J. Math. Anal. Appl., 49 (1975), 504-511. doi: doi:10.1016/0022-247X(75)90193-6.

[10]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353.

[11]

C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Differ. Equations, 1 (1989), 199-244. doi: doi:10.1007/BF01047831.

[12]

C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Varieties inertielles pour l'equation de Kuramoto-Sivashinsky. (Inertial manifolds for the Kuramoto-Sivashinsky equation), C. R. Acad. Sci., Paris, Ser. I, 301 (1985), 285-288.

[13]

A. Yu. Goritskiĭ and V. V. Chepyzhov, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 485-511; translation from Mat. Sb., 196 (2005), 23-50.

[14]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, vol. 850, Springer, 1981.

[15]

N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equations, 14 (2002), 889-941. doi: doi:10.1023/A:1020768711975.

[16]

M. A. Krasnoselski, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems, the Method of Positive Operators," Heldermann Verlag, Berlin, 1989.

[17]

Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems, 5 (1999), 233-268. doi: doi:10.3934/dcds.1999.5.233.

[18]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995.

[19]

L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, pp. 161-168, 1987.

[20]

J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866.

[21]

A. V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47. doi: doi:10.1070/IM1994v043n01ABEH001557.

[22]

R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dyn. Differ. Equations, 15 (2003), 61-86. doi: doi:10.1023/A:1026153311546.

[23]

H. Sano and N. Kunimatsu, Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems, IMA J. Math. Control Inform., 11 (1994), 75-92. doi: doi:10.1093/imamci/11.1.75.

[24]

H. Sano and N. Kunimatsu, An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems, J. Math. Anal. Appl., 196 (1995), 18-42. doi: doi:10.1006/jmaa.1995.1396.

[25]

G. R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Differ. Equations, 96 (1992), 203-255. doi: doi:10.1016/0022-0396(92)90152-D.

[26]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer, New York, 2002.

[27]

S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziaris, Order reduction for nonlinear dynamic models of distributed reacting systems, Journal of Process Control, 10 (2000), 177-184. doi: doi:10.1016/S0959-1524(99)00029-3.

[28]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997.

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